Question

A carpenter wishes to make a rain gutter with a rectangular cross-section by bending up a flat piece of metal that is 18 feet long and 44 inches wide. The top of the gutter is open. What values of x, length of metal bent up, will give a cross-sectional area of at most 50 square inches? Round your answer to three decimal places.

Answer 1

x = 20.996 , 1.002 square inches.

Explanation:

We assume that the sides are bent at right angles

Dimensions of the given metal sheet are

Length = 18feet = 216 inches

Width = 44 inches

When we will bend the width part of the given metal sheet

let the bent length be x

so the cross sectional dimensions will be

width = 44-2x

length = x

Area = x(44-2x)

The given are should be at most 50 square inches

so the equation becomes

x(44-2x)
`\leq 50`

-2x^2 + 44x
`\leq 50`

`2x^2 -44x +50\geq 0`

`x^2 -22x +25\geq 0` (dividing the equation by 2)

upon solving the above quadratic equation we get

x = 20.996 , 1.002

both the solutions are possible.

Therefore we have the are using the values of x as 20.996 and 1.002 are

42.159968 and 42.079992 which is less than 50 square inches.